Hi there!
Since we're trying to find a circular rug in a rectangular room room, we'll use a diagram like the image attached. As you can see, the circle uses the width of the room for its diameter. Since the room is 12 by 18, 12 would be the diameter of the circle since it is also the width of the room. Now, we need to find the area of the circle with a diameter of 12 feet. The radius would be 6 feet because the radius is always half of the diameter. The formula for the area of a circle is A = pi x r^2. All we need to do is to plug in the numbers to find the area of the circle. Plugging in the numbers, we get A = pi x 36. (36 is 6 squared) Therefore, the largest circular rug that can fit in James' living room is 36pi ft^2 or 113.1ft^2.
Hope this helps!! :)
If there's anything else that I can help you with, please let me know!